3.4 Newtonian equations of motion for extended bodies

Before ending this section, we present some equations for Newtonian extended bodies (stars). These
equations will give a useful guideline when we develop our formalism.

The basic equations are the equation of continuity, the Euler equation, and the Poisson equation,
respectively:

We define the mass, the dipole moment, the quadrupole moment, and the momentum of the star as

Here is a representative point of the star . The time derivative of the mass vanishes. Setting the
time derivative of the dipole moment to zero gives the velocity momentum relation and a definition of the
center of mass,

where . Using the velocity momentum relation, we calculate the time derivative of the
momentum,

where is defined by Equation (43). The Newtonian potential can be expressed by the mass and
multipole moment as